This document discusses the concepts of total angular momentum, orbital angular momentum, spin angular momentum, and their quantum numbers. It explains how the angular momenta are combined vectorially to give the total angular momentum. Spin-orbit coupling arises from the interaction of an electron's spin and orbital angular momentum with an internal magnetic field. Hund's rules describe how the total electron configurations are determined in many-electron atoms. LS and jj coupling schemes are described for combining the angular momenta in atoms.

1. Vector coupling of angular momentum

2. Total Angular Momentum
L, Lz, S, SzJ and Jz are quantized
Orbital angular momentum Spin angular momentum
Total angular momentum

3. Total Angular Momentum
• If j and mj are quantum numbers for the single electron (hydrogen
atom)
• Quantization of the magnitudes
• The total angular momentum quantum number for the single electron
can only have the values

4. The Total Angular Momentum Diagram
Figure 8.5 When forming the total angular
momentum from the orbital and spin angular
momenta, the addition must be done
vectorially, .

5. Spin-Orbit Coupling
• An effect of the spins of the electron and the orbital angular momentum
interaction is called spin-orbit coupling.
• is the magnetic field due to the proton
where cos a is the angle between
• The dipole potential energy
• The spin magnetic moment 
• .

6. Total Angular Momentum
No external magnetic field:
• Only Jz can be known because the uncertainty principle forbids Jx or Jy
from being known at the same time as Jz

7. Total Angular Momentum
With an internal magnetic field:
• will precess about

8. Total Angular Momentum
• Now the selection rules for a single-electron atom become
– Δn = anything Δℓ = ±1
– Δmj = 0, ±1 Δj = 0, ±1
• Hydrogen energy-level diagram for n = 2 and n = 3 with the spin-orbit
splitting

9. The Energy-Level
Diagram of Sodium

10. Many-Electron Atoms
Hund’s rules:
1) The total spin angular momentum S should be maximized to the extent
possible without violating the Pauli exclusion principle.
2) Insofar as rule 1 is not violated, L should also be maximized.
3) For atoms having subshells less than half full, J should be minimized.
• For labeled two-electron atom
• There are LS coupling and jj coupling to combine four angular momenta J.

11. LS Coupling
• This is used for most atoms when the magnetic field is weak.
• If two electrons are single subshell, S = 0 or 1 depending on whether the
spins are antiparallel or parallel.
• For given L, there are 2S + 1 values of J
• For L > S, J goes from L − S to L + S
• For L < S, there are fewer than 2S + 1 possible J values
• The value of 2S + 1 is the multiplicity of the state

12. LS Coupling
• The notation for a single-electron atom becomes
n2S+1 LJ
• The letters and numbers are called spectroscopic symbols.
• There are singlet states (S = 0) and triplet states (S = 1) for two electrons.

13. Energy level diagram of Mg
LS Coupling
• There are separated energy
levels according to whether
they are S = 0 or 1
• Allowed transitions must have
ΔS = 0
• No allowed (forbidden)
transitions are possible
between singlet and triplet
states with much lower
probability
• This is a two-electron atom
with one electron in the 3s
and the other excited into the
nl subshell

14. LS Coupling
• The allowed transitions for the LS coupling scheme are
– ΔL = ±1 ΔS = 0
– ΔJ = 0, ±1 (J = 0 → J = 0 is forbidden)
• A magnesium atom excited to the 3s3p triplet state has no lower
triplet state to which it can decay.
• It is called metastable, because it lives for such a long time on the
atomic scale.

15. jj Coupling
• It is for the heavier elements, where the nuclear charge causes the spin-
orbit interactions to be as strong as the force between the individual
and .
• The coupling order becomes